In probability theory, a Markov kernel (or stochastic kernel) is a map that plays the role, in the general theory of Markov processes, that the transition matrix does in the theory of Markov processes with a finite state space.[1][2]
Formal definition
Let
,
be measurable spaces. A Markov kernel with source
and target
is a map
with the following properties:
- The map
is
- measureable for every
.
- The map
is a probability measure on
for every
.
(i.e. It associates to each point
a probability measure
on
such that, for every measurable set
, the map
is measurable with respect to the
-algebra
.)
Examples
,
describes the transition rule for the random walk on
.
![{\displaystyle \kappa (x,B)={\begin{cases}\delta _{0}(B)&\quad x=0,\\P[\xi _{1}+\dots +\xi _{x}\in B]&\quad {\text{else,}}\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb788b3644731dc77e0644505fdca2fdfe4cc31)
with i.i.d. random variables
.
.
Properties
Semidirect product
Let
be a probability space and
a Markov kernel
from
to some
. Then there exists a unique
measure
on
, s.t.
.
Regular conditional distribution
Let
be a Borel space,
a
- valued random variable on the measure space
and
a sub-
-algebra.
Then there exists a Markov kernel
from
to
, s.t.
is a version of the conditional expectation
for every
, i.e.
.
It is called regular conditional distribution of
given
and is not uniquely defined.
Estimation
The kernel can be estimated using kernel density estimation.[3]
References
- {{#invoke:citation/CS1|citation
|CitationClass=citation
}}
- §36. Kernels and semigroups of kernels